CENTRALIZERS OF ELEMENTARY ABELIAN p - SUBGROUPS
AND MOD - p - COHOMOLOGY OF PROFINITE GROUPS
by
Hans-Werner Henn
1. Introduction
1.1______Let G be a profinite group and p be a fixed prime. In this paper we wi*
*ll be concerned
with H*c(G; Fp), the continuous cohomology of G with coefficients in the trivia*
*l module Fp.
We will abbreviate H*c(G; Fp) by H*(G; Fp), or simply by H*G if p is understood*
* from the
context. We recall that if G is the (inverse) limit of finite groups Githen H*G*
* = colimH*Gi.
Throughout this paper we will assume that H*G is finitely generated as Fp - alg*
*ebra. By work
of Lazard [La] it is known that this holds for many interesting groups, for exa*
*mple for profinite
p - analytic groups like GL(n; Zp), the general linear groups over the p - adic*
* integers. In case
H*G is finitely generated as Fp - algebra Quillen has shown [Q1] that there are*
* only finitely
many conjugacy classes of elementary abelian p - subgroups of G (i.e. groups is*
*omorphic to
(Z=p)n for some natural number n). In other words, the following category A(G) *
*is equivalent
to a finite category: objects of A(G) are all elementary abelian p - subgroups *
*of G, and if
E1 and E2 are elementary abelian p - subgroups of G, then the set of morphisms *
*from E1 to
E2 in A(G) consists precisely of those homomorphisms ff : E1 -! E2 of abelian g*
*roups for
which there exists an element g 2 G with ff(e) = geg-1 8e 2 E1. The category A(*
*G) plays
an important role both in Quillen's results and in the work presented here.
This category entered into Quillen's work as follows. The assignment E 7! H*E *
*extends
to a functor from the opposite category A(G)op to graded Fp - algebras and the *
*restriction
homomorphisms H*G -! H*E (for E running through the elementary abelian p - subg*
*roups
of G) induce a canonical map of algebras q : H*G -! limA(G)opH*E.
THEOREM 1.2 [Q1]. Let G be a profinite group and assume H*G is a finitely gener*
*ated Fp
- algebra. Then the canonical map q : H*G -! limA(G)opH*E is an F - isomorphis*
*m, in
other words q has the following properties.
o If x 2 Kerq, then x is nilpotent.
o If y 2 limA(G)opH*E then there exists an integer n with ypn 2 Im q.
1.3______In our main result we use the full subcategory A*(G) of A(G) whose obj*
*ects are all
elementary abelian p - subgroups except the trivial subgroup. The centralizer *
*CG (E) of
an elementary abelian p - subgroup E is a closed subgroup and hence inherits a *
*natural
profinite structure from G. The assignment E 7! H*CG (E) extends to a functor f*
*rom A*(G)
to graded Fp - algebras and the restriction homomorphisms H*G - ! H*CG (E) (for*
* E
1
running through the non-trivial elementary abelian p - subgroups of G) induce a*
* canonical
map ae : H*G -! limA*(G)H*CG (E). Our main result reads as follows.
THEOREM 1.4. Let G be a profinite group and assume H*G is a finitely generated*
* Fp
- algebra. Then the canonical map ae : H*G -! limA*(G)H*CG (E) has finite kern*
*el and
cokernel.
1.5._Remarks/Questions:______
a) The map ae of Theorem 1.4 is an actual isomorphism if G is a finite group or*
* a compact Lie
group (in the latter case H*G has to be interpreted as the mod p cohomology of *
*the classifying
space BG) provided G contains any elements of order p. This was first proved by*
* Jackowski
and McClure [JM] and then reproved and extended to certain classes of unstable *
*algebras
over the Steenrod algebra by Dwyer and Wilkerson [DW]. One's first reaction mig*
*ht be that
because the continuous cohomology of a profinite group is the colimit of the co*
*homology of
finite groups, the profinite case should be a direct consequence of the finite *
*case by passing to
appropriate (co-)limits. However, one gets confronted with a subtle problem of *
*interchanging
limits and colimits, and this has the effect that ae need not be an isomorphism*
* for a profinite
group.
In fact, our proof requires a different approach: in [He] we investigated an a*
*ppropriately
defined map ae for any unstable algebra K over the Steenrod algebra which is fi*
*nitely generated
as Fp - algebra and showed that this map has always finite kernel and cokernel *
*(see Theorem
2.5 below). In section 2 we explain this algebraic result and show how Theorem *
*1.4 can be
deduced from it. We emphasize that we do not know any proof of Theorem 1.4 whic*
*h does
not use the Steenrod algebra, in particular Lannes' T - functor in a crucial wa*
*y.
b) Obviously Theorem 1.4 gives more precise information than Theorem 1.2, but o*
*n the other
hand its applicability is more limited. For example, the major reason for worki*
*ng with A*(G)
instead of A(G) was to avoid the appearance of H*G in the limit. However, if G*
* contains
central elements of order p, then H*G does appear in the limit anyway, and Theo*
*rem 1.4 is
not very useful. In other cases the functor E 7! H*CG (E) may be too complicat*
*ed to be
evaluated. However, in these cases Theorem 1.4 may still be of some theoretical*
* interest (see
2.10 and 2.11 below for examples).
c) Theorem 1.4 says in particular that there is a least integer d such that ae *
*is an isomorphism
in cohomological degrees > d. It would be interesting to have effective upper b*
*ounds for d in
group theoretical terms. For profinite p - analytic groups the work of Lazard [*
*La] suggests
the dimension of such a group as a candidate for an upper bound for the number *
*d. In fact,
if such a group does not contain any elements of order p, as well as in the exa*
*mples discussed
in section 4 and 5 below, this gives actually a correct upper bound.
1.6______If G does not contain elements of order p then the target of ae is the*
* trivial algebra and
Theorem 1.4 says that H*G is a finite algebra. Of course, this could have also *
*been directly
deduced from Theorem 1.2.
2
However, Theorem 1.4 may already be quite interesting in case the p - rank of G*
* is equal to
one. We recall that the p - rank rp(G) of G is defined as the supremum of all n*
*atural numbers
n such that G contains an elementary abelian p - subgroup E of rank n, i.e. E ~*
*=(Z=p)n. In
case rp(G) = 1 the inverse limit simplifies substantially because A*(G) is equi*
*valent to the
following discrete category: objects are in one to one correspondence with conj*
*ugacy classes
of subgroups E ~= Z=p; the only morphisms of this category are automorphisms an*
*d the
automorphism group of an object E identifies with NG (E)=CG (E) with NG (E) den*
*oting the
normalizer of E in G. In particular we get the following corollary.
COROLLARY 1.7. Let G be a profinite group and assume H*G is a finitely generate*
*d Fp -
algebra and rp(G) = 1. Then the restriction maps induce a map
Y
ae : H*G -! (H*CG (E))NG(E)
(E)
with finite kernel and cokernel. (Here the_product is taken over conjugacy clas*
*ses of elemen-
tary abelian p - subgroups of rank 1.) |_|
Note that the group NG (E)=CG (E) is of order prime to p if E is of rank 1. The*
*refore the
invariants (H*CG (E))NG(E) are isomorphic to H*NG (E) and Corollary 1.7 can be *
*considered
as a "profinite analogue" of a result of Ken Brown on the Farell cohomology of *
*discrete groups
of p - rank 1 [B]. In this case the number d introduced in 1.5c) above correspo*
*nds to the virtual
cohomological dimension of G.
1.8.______As mentioned above section 2 will be concerned with the proof of Theo*
*rem 1.4 and related
results. In sections 3 and 4 we will study certain subgroups of the group of un*
*its in p - adic
division algebras and in section 5 we will touch upon the general linear group *
*over the p -
adic integers.
To get more explicit, we fix a prime p and a natural number n. Consider Dn, th*
*e division
algebra with invariant 1_nover the field of p - adic numbers Qp and On, the max*
*imal compact
subring of Dn. On is a local ring and reducing modulo its maximal ideal gives a*
* homomor-
phism from On to the finite field Fq with q = pn. Let Oxn denote the units of *
*On, and
Sn denote the kernel of the map Oxn- ! Fxq. In stable homotopy theory these gr*
*oups are
known as Morava stabilizer groups and their cohomology is known to play a centr*
*al role in
the chromatic theory of stable homotopy (see [M], [Ra 2,3], [D], [HG] for examp*
*le).
It is well known that rp(Sn) = rp(Oxn) 1 and equality holds iff n 0 mod (p - *
*1). Our
results give new insight if n 0 mod (p - 1). Using standard facts about divisi*
*on algebras we
determine the categories A*(Oxn) and A*(Sn) for n 0 mod (p-1) and describe the*
* structure
of the centralizers of the elementary abelian p - subgroups (Theorem 3.2.2). Fu*
*rthermore for
n = p-1 the centralizers turn out to be abelian and we can compute the target o*
*f ae explicitly,
hence H*Sn up to finite ambiguity. In particular we obtain in 3.3 the following*
* result in which
E(-) denotes an exterior algebra over Fp on the specified elements. The element*
*s yi are of
3
degree 2 and the elements xi and ai;jare of degree 1. For more details about th*
*e definition
of these elements the reader is referred to section 3.3 and 3.4.
n-1
THEOREM 1.9. Let p be an odd prime, n = p - 1 and s = p_____(p-1)2. Then the*
*re is a
homomorphism
Ys
ae : H*Sn -! Fp[yi] E(xi) E(ai;1; . .;.ai;n)
i=1
with finite kernel and cokernel.
We remark that previously the mod - p - cohomology of Sn was computed by Ravene*
*l [Ra
1] in the following cases: H1Sn and H2Sn for all n and p, all of H*Sn if either*
* n 2 and
p arbitrary or if n = 3 and p 5. So the only overlap between Ravenel's computa*
*tion and
Theorem 1.9 occurs for p = 3 and n = 2 [Ra 1, Theorem 3.3]. In section 4 we use*
* Theorem
1.9 together with some more detailed group theoretical analysis of S2 to give a*
*n independent
computation of H*Sn if p = 3 and n = 2. In thisQcase we find that ae is injecti*
*ve and we use this
to describe H*S2 as an explicit subalgebra of 2i=1F3[yi]E(xi)E(ai;1; ai;2) (T*
*heorem 4.2).
The multiplicative structure of the result derived here differs from that of Ra*
*venel although
additively the two results agree. Ravenel has informed me that he now believes *
*that there is
a mistake in his calculation. Finally we remark that Gorbounov, Siegel and Symo*
*nds [GSS]
have independently and with very different methods confirmed the calculation in*
* Theorem
4.2.
The calculations of H*S2 for primes p > 3 have been used by Shimomura and Yabe *
*[SY] to
determine the stable homotopy groups of L2S0p, the second stage in the chromati*
*c tower of the
p - local sphere. The computation of H*S2 at the prime 3 will be relevant for u*
*nderstanding
L2S0pif p = 3 and for this reason we have decided to give a rather detailed pre*
*sentation in
section 4. In fact, Shimomura [S] has already used the corrected computation o*
*f H*S2 to
compute the homotopy groups of the L2 - localization of the Toda - Smith comple*
*x V (1)
at the prime 3 up to a certain ambiguity; this ambiguity will be settled in joi*
*nt work with
Mahowald [HM] using the approach towards H*S2 via centralizers of elementary ab*
*elian p -
subgroups that we introduce in this paper.
1.10.______Our second application concerns the mod - p - cohomology of the gene*
*ral linear groups
GL(n; Zp). The following result should be compared with Ash's computations [A]*
* of the
Farrell cohomology of GL(n; Z). As above the element y has degree 2 while all o*
*ther elements
are of degree 1.
THEOREM 1.10. Let p be an odd prime and n = p - 1. Then there is a homomorphism
ae : H*GL(n; Zp) -! (Fp[y] E(x) E(a1; . .;.an))Z=n
with finite kernel and cokernel. (Here (-)Z=n denotes the invariants with resp*
*ect to the
following action of Z=n by algebra homomorphisms. After choosing a suitable iso*
*morphism
4
o : Z=n - ! Fxp this action is given by gy = o(g)y, gx = o(g)x and gaj = o(g)ja*
*j for
j = 1; : :;:n if g 2 Z=n.)
This result looks very similar to 1.9. In fact, the similarity becomes even st*
*ronger if one
compares (for n = p - 1) the groups Oxnand GL(n; Zp). In this case the targets *
*of the two
maps ae agree (cf. Theorem 3.2.2 and Theorem 5.2).
As in the case of Sn we give the complete computation for p = 3 and n = 2 (Prop*
*osition 5.5).
Acknowledgements:_______The author was partially supported by a Heisenberg fell*
*owship of the
Deutsche Forschungsgemeinschaft. The origin of this paper can be traced back t*
*o a one
month visit to Aarhus in the fall of 1990. This work was then pushed further du*
*ring a three
months visit at the Mittag Leffler institute in the fall of 1993. I would like *
*to thank the staff
at both institutions for providing such pleasant and stimulating working condit*
*ions. It is a
pleasure to express special thanks to Bob Oliver on both occassions.
2. The proof of Theorem 1.4 and related results
As already indicated in the introduction, Theorem 1.4 is deduced from a general*
* result about
certain unstable algebras over the Steenrod algebra A [He]. We will begin by re*
*calling some
facts about Lannes' T functor which are necessary to explain the main algebraic*
* result of
[He] and to deduce Theorem 1.4 from it.
2.1.______Let U resp. K denote the category of unstable modules resp. unstable *
*algebras over the
mod p Steenrod algebra A (see [L1]). The cohomology of a space is an unstable a*
*lgebra, in
particular the cohomology of any finite group and then also the cohomology of a*
*ny profinite
group is such an algebra. The Steenrod algebra is actually a Hopf algebra and i*
*ts diagonal
gives rise to a tensor product on the categories U resp. K.
Now let V be an elementary abelian p - group with mod p cohomology H*V . Lanne*
*s [L1]
has introduced the functor TV : U - ! U. It is left adjoint to tensoring with*
* H*V , i.e
Hom U(TV M; N) ~= Hom U (M; H*V N) for all unstable modules M and N. TV has*
* a
number of remarkable properties. In particular, TV lifts to a functor from K *
*to itself and
the adjunction relation continues to hold in K: Hom K (TV K; L) ~=Hom K (K; H*V*
* L) for all
unstable algebras K and L.
2.2.______Now let G be a finite group. The following computation of TV H*G in [*
*L2] (for a more
accessible reference see also [L1,3.4]) is quite crucial for the proof of Theor*
*em 1.4.
Denote by Rep (V; G) the set of G - conjugacy classes of homomorphisms from V t*
*o G. For
each conjugacy class choose a representative ' and denote the centralizer of Im*
* ' in G by
CG ('). The homomorphism c' : V x CG (') ! G, (v; g) 7! v'(g) induces a map of *
*unstable
algebras c'* : H*G -! H*V H*CG (') which is adjoint to a map of unstable algeb*
*ras
ad(c'*) : TV H*G -! H*CG (').
5
THEOREM 2.2 [L2]. The homomorphism of unstable algebras
Y
TV H*G -! H*CG (')
'2Rep(V;G)
*
* _
whose components are the maps ad(c'*) is an isomorphism for each finite group G*
*. |_|
Note that the theorem shows in particular that the natural map from Rep (V; G) *
*to
Hom K(H*G; H*V ), which sends ' to its induced map '*, is a bijection.
We also see that the terms in the inverse limit occuring in Theorem 1.4 appear *
*in the com-
putation of TV H*G. Dwyer and Wilkerson [DW] noticed that this allows a purely *
*algebraic
approach to the map ae of Theorem 1.4 which makes sense for a much larger class*
* of unstable
algebras. In order to explain this we need some more preparations (see [DW] and*
* [HLS, I.4
and I.5]).
2.3.______To an unstable algebra K we associate a category R(K) as follows. Its*
* objects are the
morphisms of unstable algebras ' : K ! H*V , V an elementary abelian p - group,*
* for which
H*V becomes a finitely generated K - module via '; sometimes it will be conven*
*ient to
denote such an object by the pair (V; '). Then the set of morphisms from (V1; '*
*1) to (V2; '2)
are all homomorphisms V1 -ff!V2 of abelian groups such that '1 = ff*'2. If K is*
* noetherian
the opposite of this category was first investigated by Rector [Rc]. The full s*
*ubcategory of
R(K) having as objects all (V; ') with V non-trivial will be denoted by R*(K)
If K = H*BG then R(K) is equivalent to Quillen's category A(G) and R*(K) is equ*
*ivalent
to A*(G). In fact, in this case the computation of Hom K (H*G; H*V ) (see 2.2 a*
*bove) can be
used to identify the objects of R(K) with the monomorphic representations of el*
*ementary
abelian p - groups in G and an equivalence between R(K) and a skeleton of A(G) *
*is induced
by associating to a homomorphism ' : V - ! G the unique object in the skeleton *
*of A(G)
which is isomorphic to the image of ' (see [HLS, I.5.3]).
2.4.______Now consider the unstable algebra TV K. For a morphisn ' : K - ! H*V*
* we obtain
a connected component TV (K; ') of TV K: it is defined as TV (K; ') := Fp(') T0*
*VK TV K
where Fp(') denotes Fp considered as a module over TV0K via the adjoint of '.
More generally, we can consider the category K - U whose objects are unstable A*
* - modules
M with A - linear K - module structure maps K M -! M, and whose morphisms are *
*all
A - linear maps which are also K - linear. The full subcategory of K - U consis*
*ting of those
objects which are finitely generated as K - modules is denoted by Kfg - U.
If M is in K - U then TV M is in TV K - U and one can define components TV (M; *
*') :=
Fp(') T0VK TV M which are modules over the corresponding components TV (K; '). *
* From
the adjoint of the identity of TV K we obtain canonical algebra morphisms aeK;(*
*V;')from
K to TV (K; '), hence TV (M; ') can be considered as a K - module and the assig*
*nment
(V; ') 7! TV (M; ') gives rise to a functor from R(K) to K - U. Furthermore th*
*e adjoint
of the identity of TV M gives rise to maps aeM;(V;')from M to TV (M; ') which a*
*re all K -
6
linear, and we obtain a natural transformation between the constant functor fro*
*m R(K) to
K - U with value M and the functor (V; ') 7! TV (M; ').
If G is a finite group and K = M = H*G then Theorem 2.2 implies that the func*
*tor
(V; ') 7! TV (M; ') corresponds via the equivalence between R(K) and A(G) to th*
*e functor
E 7! H*CG (E) that appears in Theorem 1.4. Furthermore, the maps aeM;(V;')corre*
*spond to
the restriction maps H*G -! H*CG (').
Now we are ready to formulate the general algebraic theorem from which Theorem *
*1.4 is
deduced.
THEOREM 2.5 [He, Cor. 3.10]. Let K be a noetherian unstable algebra and let M*
* be an
object in Kfg - U. Then the maps aeM;(V;')induce a map
ae : M -! limR*(K)TV (M; ')
_
which has finite kernel and cokernel. |_|
In fact, it is shown in [He] that this map is localization away from finite obj*
*ects in Kfg - U.
Theorem 2.5 together with Theorem 2.2 yield Theorem 1.4 in the case of a finite*
* group (in
which case ae is even an isomorphism by [JM] and [DW]). To prove Theorem 1.4 fo*
*r profinite
groups G for which H*G is a finitely generated Fp - algebra, it suffices to ext*
*end Theorem
2.2 to this setting. We recall that a profinite group G is given as the (invers*
*e) limit limiGi of
finite groups Gi along a directed partially ordered set (I; ) which we think of*
* as a category
(denoted I for simplicity) in the usual way. Then H*G can be identified with co*
*limiH*Gi.
Here is the extension of Theorem 2.2.
THEOREM 2.6. Assume G is a profinite group such that H*G is a finitely generate*
*d Fp -
algebra. Then the homomorphism of unstable algebras
Y
TV H*G -! H*CG (')
'2Rep(V;G)
whose components are the maps ad(c*') is an isomorphism.
In this result the set Rep (V; G) of representations is defined as before, i.e.*
* the topology on
G does not play any role. However, the centralizer of an elementary abelian p -*
* subgroup of
a profinite group inherits a natural structure of a profinite group and this st*
*ructure is used
in 2.6.
Proof_of_2.6:_____We deduce Theorem 2.6 from Theorem 2.2. For this we note that*
* TV commutes
with arbitrary colimits, i.e. we get
Y
TV H*G = colimiTV H*Gi ~=colimi H*CGi(') :
'2Rep(V;Gi)
7
Let us look more carefully at the maps in the inverse system. For a morphism *
*: i ! j
in I we use the same letter for the associated maps from Gj to Gi and from CGj(*
*') to
CGi(') (with ' 2 Rep (V; Gj)). If we identify TV H*Gi as in 2.2 then the map T*
*V * :
TV H*Gi -! TV H*Gj isQgiven as follows: the ' - th component of it (' 2 Rep(V; *
*Gj)) sends
the family {x'0} 2 '02Rep(V;Gi)H*CGi('0) to the element *(x' ) 2 H*CGj('). He*
*re *
is the inducedQmap from H*CGi(') to H*CGj(').Q In fact, in the same way we get *
*maps
TV ss*ifrom '02Rep(V;Gi)H*CGi('0) to '2Rep(V;G)H*CG (') if ssi denotes the *
*canonical
map from G to Gi. These maps fit together and thus give a map
Y Y
colimi H*CGi('0) -! H*CG (')
'02Rep(V;Gi) '2Rep(V;G)
which we denote TV ss* by abuse of notation and which we claim to be an isomorp*
*hism. In
order to show this we need the following lemmas.
LEMMA 2.7. Let G = limGi be any profinite group.
a) Then the maps ssi : G -! Gi induce a bijection Rep(V; G) -! limiRep(V; Gi) o*
*f profinite
sets for each elementary abelian p - group V .
b) For any homomorphism ' from an elementary abelian p - group V into G the ma*
*ps
ssi : G -! Gi induce an isomorphism CG (') -! limiCGi(ssi') of profinite groups.
LEMMA 2.8. Let G be any profinite group for which H*G is a finitely generated F*
*p - algebra.
Then the set Rep (V; G) is finite for each elementary abelian p - group V .
We postpone the proofs of 2.7 and 2.8 and continue with the proof of 2.6.
Q
First we show that TV ss* is onto. So let x = {x'} 2 '2Rep(V;G)H*CG (') be g*
*iven. For
each ' 2 Rep(V; G) we find by 2.7.b) an object i = i(') of I and an element yi *
*2 H*CGi(ssi')
such that x' = (ssi)*yi. By 2.8 there are only finitely many ' and therefore we*
* can assume
that i is independent of '. Furthermore, by 2.7.a) and 2.8 we can choose i suc*
*h that in
addition the natural map Rep (V; G) ! Rep (V; Gi) is injective. For such an i a*
*nd any '0 2
Rep (V; Gi) let z'0 2 H*CGi('0) be equal to yi(')if '0 = ssi' for some (necessa*
*rily unique)
', and chooseQarbitrarily elements z'0 if there is no ' such that '0= ssi'. The*
*n the family
z = {z'0} 2 '02Rep(V;Gi)H*CGi('0) satisfies TV ss*i(z) = x, and hence TV ss* *
*is onto.
Q
To see that TV ss* is mono we show that for any element z = {z'0} 2 '02Rep(V;*
*Gi)H*CGi('0)
with TV ss*(z) = 0 there is : i -! j in I such that TV *(z) = 0. Now TV ss*(z)*
* = 0 implies by
2.7.b) that for each '0there is '0 : i -! j('0) such that for each lift ' of '0*
*to Rep(V; Gj('0))
which further lifts to Rep (V; G) we get ('0)*(z'0) = 0 in H*CGj('0)('). As the*
*re are only
finitely many '0 we can choose a common : i -! j such that *(z'0) = 0 in H*CGj*
*(') for
each '0and each lift ' of '0to Rep(V; Gj) which further lifts to Rep(V; G). Fur*
*thermore by
2.7.a) we can find 0 : j -! j0 such that each element ' 2 Rep (V; Gj) which doe*
*s not lift
to Rep (V; G) does_not lift to Rep (V; Gj0) either. If = 0 : i -! j0 then it *
*is clear that
TV *(z) = 0. |_|
8
Proof_of_2.7:_____The proof is an exercise in elementary point set topology. We*
* sketch part (a)
and leave part (b) to the reader.
Let us first show surjectivity. So assume we have a compatible family of eleme*
*nts 'i 2
RepQ(V; Gi). For a subset J of I let Rep J be the set of families of homomorphi*
*sms {OEi} 2
i2IHom (V; Gi) such that the conjugacy class of OEi is equal to 'i whenever *
*i 2 J and such
that the OEi are compatible as long as i 2 J . We have to show that Rep I is n*
*on-empty.
Because any two elements of I have an upper bound it is clear that Rep J is non*
*-empty
whenever J is finite. Furthermore Rep J is easily seen to be closed for any J .*
* Now we use
that the intersection of closed sets in a compact space is non-empty if every f*
*inite intersection
is non-empty and we are done.
Next assume that we have two elements ' and '0in Rep(V; G) represented by homom*
*orphisms
OE and OE0 such that ssiOE and ssiOE0 are conjugate for each i, i.e. there is *
*an element gi 2 Gi
with ssiOE(v) = gissiOE0(v)g-1ifor all v 2 V . For injectivity in (a) it is eno*
*ugh to show that the
family gi can be chosen to be compatible. Again this is easy for any finite sub*
*set of I, and
the general case follows again from the fact that an intersection_of closed set*
*s in a compact
space is non-empty if every finite intersection is non-empty. |_|
Proof_of_2.8:_____For any profinite G we have by 2.2 and 2.7.a)
Rep (V; G) ~=limiRep(V; Gi) ~=limiHom K (H*Gi; H*V ) :
Furthermore we can identify limiHom K (H*Gi; H*V ) with Hom K (H*G; H*V ). Fin*
*ally, if
H*G is a finitely generated_Fp - algebra, then the set Hom K (H*G; H*V ) is cle*
*arly finite and
hence we are done. |_|
2.9.______We would like to point out that Theorem 2.2 can be used to derive num*
*erous qualitative
results on cohomology of finite groups such as detection results, information o*
*n nilpotent
elements in H*G or characterizations of the "support of elements" in H*G (cf. [*
*HLS, I.5],
[CH]). Because of Theorem 2.6 all these results have anologues for continuous c*
*ohomology of
profinite groups G as long as H*G is a finitely generated Fp - algebra. We list*
* here only the
following result which is a special case of the profinite analogue of Theorem 2*
* of [CH].
PROPOSITION 2.10. Let G be profinite with p - rank one and assume H*G is finit*
*ely
generated as Fp - algebra. Then the map
Y
ae : H*G -! (H*CG (E))NG(E)
(E)
(cf. Corollary 1.7) is a monomorphism if and only if H*G is free over a polynom*
*ial subalgebra
of H*G with one generator.
Proof:______Assume that H*G is free over a polynomial subalgebra on one generat*
*or. Then the
same is true for any non-trivial ideal in H*G, in particular for the kernel of *
*the map ae.
However, by Corollary 1.7 this ideal is finite, hence it must be trivial.
9
Conversely, assume that ae is a monomorphism. Take any element x in H*G which r*
*estricts
to a non-nilpotent element on all non-trivial elementary abelian p subgroups of*
* rank 1. Such
an x exists by Theorem 1.2. We can consider the cohomology of H*CG (E) as a mod*
*ule over
the polynomial subalgebra of H*G generated by x (again via a restriction homomo*
*rphism).
Now the proof of Theorem 1.1 in [BC] (see also Remark 2.3 in the same paper) sh*
*ows that
H*CG (E) is freeQover this polynomial subalgebra. By assumption_H*G is a submo*
*dule of
the free module (E)H*CG (E) and hence it is also free. |_|
2.11.______Finally we want to point out that Theorem 1.4 can also be used to ge*
*t information on
cohomology with non-trivial coefficients. For example, assume M is a finite con*
*tinuous G -
module with a composition series for which all successive subquotients are triv*
*ial modules.
(Such composition series exist alwaysQif G is a pro - p - group.) Furthermore a*
*ssume that the p
- rank of G is one and ae : H*G -! (E)(H*CG (E))NG(E) is an isomorphism in de*
*grees > d.
Then playing with the long exact sequences in cohomology associatedQto short ex*
*act sequences
of coefficient modules shows that the map ae : H*(G; M) -! (E)(H*(CG (E); M))*
*NG(E) is
an isomorphism in all degrees > d + 1.
3. The case of the stabilizer groups
3.1.______In this section we will apply our general results to certain subgroup*
*s of p - adic division
algebras which play an important role in stable homotopy theory. We begin by re*
*calling the
definition and basic properties of these groups. The reader is referred to [Rn,*
* Chap. 3,7] and
[Ha, 20.2.16, 23.1.4] for background information on division algebras.
3.1.1._____Let p be a prime. For each integer n let Wn be the ring of Witt vect*
*ors of the finite
field Fq with q = pn elements and let oe : Wn ! Wn, w 7! woebe the lift of the *
*Frobenius
automorphism x 7! xp on Fq. Adjoin an element S to Wn subject to the relations *
*Sn = p,
Sw = woeS for each w 2 Wn. The resulting non-commutative ring will be denoted b*
*y On. It
is the maximal order in the central division algebra Dn over the field Qp of ra*
*tional numbers
with invariant 1=n and is a free module over Wn of rank n with generators the e*
*lements
Si; 0 i < n. An important property of Dn that we will use below is that the de*
*gree (over
Qp) of each commutative subfield of Dn divides n, and each extension of Qp whos*
*e degree
divides n can be embedded as commutative subfield of Dn.
We recall that On can be identified with the endomorphism ring of a certain for*
*mal group law
over Fq of height n [Ha, Theorem 20.2.13]. Its group of units Oxnis often calle*
*d the n - th (full)
Morava stabilizer group. The element S generates a two sided maximal ideal m in*
* On with
quotient On=m = Fq. The kernel of the resulting epimorphism of groups Oxn- ! (F*
*q)x will
be denoted by Sn and is also called the (strict) Morava stabilizer group; it ca*
*n be identified
with the group of strict automorphisms of the same height n formal group law ov*
*er Fq.
3.1.2._____The groups Oxn and Sn have natural profinite structures which can be*
* described as
follows. The valuation v on Qp (normalized such that v(p) = 1) extends uniquel*
*y to a
10
valuation on Dn such that v(S) = 1_nand On = {x 2 Dn|v(x) 0}. The two sided ma*
*ximal
ideal m is given by m = {x 2 Dn|v(x) > 0}. The valuation gives subgroups
FiSn := {x 2 Sn|v(1 - x) i} = {x 2 Sn|x 1 mod Sni}
for positive multiples i of 1_nwith
Sn = F_1nSn F_2nSn : :::
The intersection of all these subgroups is empty and Sn is complete with respec*
*t to this
filtration, i.e. we have Sn = limiSn=FiSn. Furthermore, we have canonical isomo*
*rphisms
FiSn=Fi+_1nSn ~=Fq
induced by
x = 1 + aSi 7! a :
Here a is an element in On, i.e. x 2 FiSn and a is the residue class of a in O*
*n=m = Fq
In particular, all the quotients Sn=FiSn are finite p - groups and hence Sn is *
*a profinite p -
group which is the p - Sylow subgroup of the profinite group Oxn.
3.1.3._____The associated graded object grSn with griSn = FiSn=Fi+_1nSn becomes*
* a graded Lie
algebra with Lie bracket [a; b] induced by the commutator xyx-1y-1 in Sn. Furt*
*hermore,
if we define a function ' from the positive real numbers to itself by '(i) := m*
*in{i + 1; pi}
then the p - th power map on Sn induces maps P : griSn -! gr'(i)Sn which define*
* on grSn
the structure of a mixed Lie algebra in the sense of Lazard [La, Chap. II.1]. I*
*f we identify
the filtration quotients with Fq as in 3.1.2 above then the Lie bracket and the*
* map P are
explicitly given as follows.
LEMMA 3.1.4.
a) Let a 2 griSn, b2 grjSn. Then
ni pnj
[a; b] = abp - ba 2 gri+jSn :
b) Let a 2 griSn. Then
8 ni (p-1)ni
< a1+p +:::+p if i < (p - 1)-1
P a= : a + a1+pni+:::+p(p-1)niif i = (p - 1)-1
a if i > (p - 1)-1 .
Proof:______a) Write i = k_n, j = l_nand choose representatives x = 1 + aSk 2 F*
*iSn, y = 1 + bSl 2
FjSn. Then x-1 = 1 - aSk mod Sk+1, y-1 = 1 - bSl mod Sl+1 and the formula
xyx-1y-1 = 1 + ((x - 1)(y - 1) - (y - 1)(x - 1))x-1y-1
11
shows
xyx-1y-1 = 1 + (aSkbSl- bSlaSk) mod Sk+l+1 :
Because On=m ~= Wn=(p) we can choose a and b from Wn. Then Sw = woeS and woe
wp mod (p) give the stated formula.
b) Again we write i =Pk_nand choose a representative x = 1 + aSk with a 2 Wn. C*
*onsider
the expression xp = r pr(aSk)r. Because pris divisible by p for 0 < r < p an*
*d because
Sn = p we get
xp 1 + aSn+k + : :+:(aSk)p mod S2k+n :
Furthermore
k oe(p-1)kpk pk p(p-1)kpk 1+pk+:::+p(p-1)kpk pk+1
(aSk)p = aaoe : :a: S aa : :a: S a S mod S *
* :
Now we only have to determine whether pk is smaller resp. equal resp. larger th*
*an n + k.
i.e. whether pi is smaller resp. equal resp. larger than 1 + i. These cases are*
*_equivalent to
i < (p - 1)-1 resp. i = (p - 1)-1 resp. i > (p - 1)-1 and hence we are done. |_|
Remark:_______One can use 3.1.4 to compute H1(FiSn) resp. H1(FiSn) (coefficien*
*ts are, as al-
ways, in Fp). For example, by using [Lazard, III.2.1, in particular III.2.1.8]*
* we can derive
the following. If i > (p - 1)-1 then the quotient map FiSn -! FiSn=Fi+1Sn indu*
*ces an
isomorphism on H1. Furthermore, if i 1 then FiSn=Fi+1Sn is elementary abelian *
*of rank
n2 and, if p is odd and i 1, then H*FiSn is an exterior algebra on H1FiSn (see*
* [Lazard,
V.2.2.7]). Ravenel claims in [Ra 2, Theorem 6.3.7] that H*FiSn is exterior on n*
*2 generators in
dimension 1 as soon as i > __p___2(p-1). (Note that Ravenel's i corresponds to *
*i_nin our notation!)
However, if p = 5, n = 4 and i = 3_4it is not hard to show (using 3.1.4) that t*
*he abelianization
of FiSn=Fi+1Sn is FiSn=Fi+n-1_nSn which is elementary abelian of rank 12. Henc*
*e H1FiSn
and H1FiSn are also of dimension 12 only.
3.2______The algebras H*(Oxn) and H*Sn are known to be finitely generated Fp - *
*algebras (e.g
because they have a finite index normal subgroup, say F1Sn, whose cohomology al*
*gebra is
even finite), hence Theorem 1.4 and its consequences can be applied to both gro*
*ups. In order
to do this we need to determine the categories A*(G), G = Oxnor G = Sn. The fir*
*st step to
determine these categories is given by the following well known theorem. For th*
*e convenience
of the reader we repeat its short proof.
THEOREM 3.2.1. The groups Oxnresp. Sn have elements of order p iff n 0 mod (p *
*- 1)
in which case both groups have p - rank 1.
Proof:______If A is any finite abelian subgroup of Dxnthen A generates a commut*
*ative subfield K
of Dn and A is a finite subgroup of its roots of unity. However, for any commut*
*ative field the
roots of unity form a cyclic subgroup and hence the p - rank is at most 1. Furt*
*hermore, the
p - rank of the units Dxnis 1 if and only if Dn contains the cyclotomic field Q*
*p(ip) of degree
12
p - 1 over Qp which happens if and only if n is a multiple of p - 1. Finally an*
*y element of
finite order in Dxnmust have valuation 0, i.e. is contained in Oxn; furthermore*
*,_if the order of
the element is a power of p it must be in the p - Sylow subgroup Sn. |_|
Remark:_______Lemma 3.1.4.b) implies directly that an element of order p can ex*
*ist in Sn only if
1 ______ k_
p-1 is of the form n , i.e. if n = k(p - 1). In fact, in this case any non-tri*
*vialkelement(xpof-1)k
order p is necessarily contained in F_1_p-1Sn and x = 1 + aSk satisfies a+ a1+p*
* +:::+p = 0
with a 6= 0. One can show that for any a 6= 0 satisfying a+ a1+pk+:::+p(p-1)k= *
*0 one can find
a 2 On such that 1 + aSk is of order p.
The following more precise result gives the complete description of the categor*
*ies A*(G) and
of the centralizers of each object.
THEOREM 3.2.2. Let n = k(p - 1).
a) Any two subgroups of Oxn which are isomorphic to Z=p are conjugate in Oxn an*
*d each
abstract automorphism of such a subgroup E is induced by conjugation by an elem*
*ent in Oxn,
i.e. AutA*(Oxn)(E) ~= Aut(E) ~= Z=(p - 1). Furthermore the centralizer COxn(E*
*) is given
as the group of units in the maximal order in the central division algebra over*
* Qp(ip) of
dimension k2 and invariant 1_k.
n-1
b) There are exactly ___p______(p-1)(pk-1)conjugacy classes of subgroups isomor*
*phic to Z=p in Sn and
for each such subgroup E the group Aut A*(Sn)(E) is trivial. Furthermore the c*
*entralizer
CSn(E) is normal in COxn(E) with cyclic quotient of order pk - 1.
Proof:______a) First note that the group of field automorphisms of Qp(ip) maps *
*via restriction
isomorphically to the group of all abstract automorphisms of the multiplicative*
* subgroup of
order p generated by ip. Furthermore, by the Skolem - Noether Theorem any two e*
*mbeddings
'; '0 of Qp(ip) in Dn are conjugate, i.e. there exists u 2 Dxnsuch that '(x) = *
*u'0(x)u-1 for
each x 2 Qp(ip). These two facts imply immediately that the assertion on conjug*
*acy classes
and automorphisms in part a) hold if Oxnis replaced by Dxn. To show them for Ox*
*nit suffices
therefore to show that in the Skolem - Noether Theorem one can take u of valuat*
*ion 0. To
see this it is enough to note that the valuation on Dn and its restriction to t*
*he centralizer
CDn ('(Qp(ip))) takes the same values. Now CDn ('(Qp(ip))) is again a division *
*algebra which
is a central division algebra over Qp(ip) of dimension k2 and of invariant 1_k(*
*s. [Ha, 20.2.16,
23.1.4]). Then the required property of the value groups follows easily from Th*
*eorem 14.3 in
[Rn].
If E ~=Z=p Oxnthen E generates a subfield which we can identify with Qp(ip). F*
*urthermore
COxn(E) is just the intersection Oxn\ CDn (E) ~= Oxn\ CDn (Qp(ip)) which is pre*
*cisely the
group of units in the maximal order of CDn (Qp(ip)). Thus the proof of a) is co*
*mplete.
b) First we observe that for any E ~= Z=p in Sn the group Aut A*(Sn)(E) is isom*
*orphic to
NSn(E)=CSn(E), so it is a subquotient of a profinite p - group and hence a p - *
*group. However,
the abstract isomorphism group is of order p - 1, i.e. of order prime to p, so*
* we see that
Aut A*(Sn)(E) is trivial.
13
The remaining parts of b) are now deduced from a). We know from (a) that Fxqact*
*s transi-
tively on the set of Sn - conjugacy classes of subgroups E ~=Z=p. So in order t*
*o determine
the number of Sn - conjugacy classes it suffices to show that the order of the *
*isotropy group
of this action is (p - 1)(pk - 1). Now the isotropy subgroup of E is equal to t*
*he image of the
normalizer NOxn(E) under the quotient map Oxn- ! Fxq. The image of the centrali*
*zer is of
order pk - 1 because CDn (E) is of dimension k2 over Qp(ip) and hence the resid*
*ue field of
the maximal order of CDn (E) has order pk. Again by a) the quotient NOxn(E)=COx*
*n(E) has
order p - 1. We claim that the image of the COxn(E) in Fxqcontinues to have ind*
*ex p - 1 in
the image of NOxn(E) and therefore the isotropy group has order as claimed.
Assume the image of COxn(E) in Fxqhad index less than p - 1 in the image of NOx*
*n(E). By a)
again there would be a nontrivial automorphism ff of E induced by conjugation b*
*y an element
y 2 NOxn(E), and y = zx with z 2 COxn(E) and x 2 Sn. However, then we would eve*
*n have
x 2 Sn \ NOxn(E) = NSn(E) = CSn(E), hence y 2 COxn(E) and this is in contradict*
*ion to
the non-triviality of ff.
Finally, CSn(E) is normal in COxn(E) with cyclic quotient of order pk-1 because*
* CSn(E) is
the kernel of the surjective map_from the units in COn (E) to the units in the *
*residue field of
COn (E) which is of order pk. |_|
Remark:_______We remark that in case p = 2 Theorem 3.2.2 becomes trivial becaus*
*e there is a unique
central element of order 2, namely the element -1 2 On. However, in this case P*
*roposition
2.10 implies that H*Sn is finitely generated and free over a polynomial subalge*
*bra on one
generator.
If p is odd, the number of conjugacy classes grows quickly with p and n, e.g. i*
*f p = 3 and
n = 2 we get 2 classes, and if p = 5 and n = 4 we get 39 classes. We will see i*
*n the remark
after the proof of 4.3 below that for p > 3 and n = p - 1 the map ae of 2.10 is*
* not mono in
dimension 2 and hence H*Sn cannot be free over a polynomial subalgebra on one g*
*enerator
as it was claimed in [Ra 2, 6.2.10b)].
3.3.______We will now consider our main theorem for Sn in case n = p - 1 and we*
* will assume that
p is odd, the case p = 2 being trivial.
Let E be a cyclic subgroup of order p in Sn. By 3.2.2 we know that COxn(E) is g*
*iven as the
group of units in the maximal order of Qp(ip), which is equal to Zp[ip]; in par*
*ticular COxn(E)
is abelian. In fact, the units Zp[ip]x are well known to be (non-canonically) i*
*somorphic to
Z=p x (Zp)n x Z=n and hence 3.2.2 yields CSn(E) ~=Z=p x (Zp)n (see also the rem*
*ark after
3.4 below). So the cohomology of CSn(E) is isomorphic to H*(Z=p) H*(Zp)n and*
* can
be written as Fp[y] E(x) E(a1; . .;.an) where y has degree 2 and x and the el*
*ements aj
have degree 1. The following result (Theorem 1.9 of the introduction) is now an*
* immediate
consequence of Corollary 1.7 and of Theorem 3.3.2.
n-1
THEOREM 3.3. Let p be an odd prime and n = p - 1. Then Sn has _p____(p-1)2con*
*jugacy
classes of subgroups which are isomorphic to Z=p and whose centralizers are all*
* isomorphic
14
n
to Z=px(Zp)n. Choose representatives Ei, i = 1; : :;:p_-1__(p-1)2from each conj*
*ugacy class. Then
the map
Y Y
ae : H*Sn -! H*CSn(Ei) ~= Fp[yi] E(xi) E(ai;1; . .;.ai;n)
i i
induced by the restriction maps has finite kernel and cokernel.
3.4.______Now assume again that n = k(p - 1) with k arbitrary. The conjugation*
* action of Oxn
on Sn induces an action of (Fq)x on H*SnQwhich is important in applications in *
*homotopy
theory. The group (Fq)x acts also on (E)H*CSn(E) in such a way that the map a*
*e is (Fq)x
- linear. This action can be described as follows: Let E1 be a fixed cyclic sub*
*group of Sn of
order p. The action of NOxn(E1) on H*CSn(E1) inducesQan action of the image of *
*NOxn(E1)
in (Fq)x . We denote this image by N . The product (E)H*CSn(E) can be identif*
*ied with
the representation of (Fq)x which is induced from that of N on H*CSn(E1). With *
*this Fxq-
action on its target the map ae is linear.
The following result explicitly describes the action of N onQCSn(E1) in case n*
* = p - 1, and
hence gives an explicit description of the action of Fxqon (E)H*CSn(E) in thi*
*s case. Note
that in case n = (p - 1) we have isomorphisms N ~= Z=n2 and CSn(E1) ~=Z=p x (Zp*
*)n.
PROPOSITION 3.4. Let n = p - 1. The action of N ~= Z=n2 on CSn(E1) factors thro*
*ugh an
action of the quotient Z=n. As a Z=n - module CSn(E1) ~=Z=p x (Zp)n splits as t*
*he direct
sum of the module Z=p (with the natural action of Z=n ~= Aut(Z=p)) and the n di*
*fferent
one-dimensional representation of Z=n over the ring Zp.
Proof:______The first statement follows because the image C of COxn(E1) (which *
*is isomorphic to
Z=n) acts clearly trivially and hence the action factors through N =C ~=Z=n. *
*This action
agrees by the Skolem - Noether Theorem with the action of the Galois group_of t*
*he cyclotomic
extension which is well understood in number theory (see [W. p.301]). |_|
Remark:_______By Proposition 3.4 it is clear that for a suitable choice of an i*
*somorphism o : Z=n -!
Fxpand of elements y, x and aj the action of Z=n on H*(Z=p x (Zp)n) ~= Fp[y] E*
*(x)
E(a1; . .;.an) is described by the formula given in Theorem 1.10 of the introdu*
*ction. In the
next section we will have to be even more specific with the choice of these gen*
*erators so we
take the time now to explain this.
The valuation on Dn restricts to one on CDn (E) and as in 3.1.2 we get a filtra*
*tion on the
group CSn(E). If we identify CDn (E) with Qp(ip) then the maximal ideal in the*
* maximal
order of CDn (E) is generated by the element ip - 1 of valuation _1_p-1. Using *
*the p - th power
map on the associated graded of this filtration one sees that a minimal set of *
*topological
generators for CSn(E) is given by the element ip of order p and any choice of e*
*lements jj,
j = 2; : :;:n + 1 with the property that jj 1 + (ip - 1)jmod (ip - 1)j+1. Furt*
*hermore the
filtration is Z=n - invariant and because n is prime to p the elements jj can b*
*e chosen to
15
generate the different one dimensional representations_of Z=n over Zp. With su*
*ch a choice
mod p reduction gives a set of generators ip, __j2; : :;:__jn+1of H1(Zp[ip]x ).*
* If we take for x,
a2; : :;:an; an+1 =: a1 the dual basis in H1 and for y the Bockstein of x then *
*the formula
given in 1.10 holds.
4. The case p = 3 and n = 2
4.1.______In this section we will consider the case p = 3 and n = 2 in fair det*
*ail. This is the first
non-trivial case where our main theorem can be applied to get information on H**
*Sn. By
Theorem 3.3. we find two conjugacy classes of Z=3's in S2 whose centralizers ar*
*e isomorphic
to Z=3 x (Z3)2. We will compute H*S2, in particular we will showQthat the map a*
*e of 2.10
is a monomorphism and we will describe H*S2 as a subalgebra of 2i=1F3[yi] E(*
*xi)
E(ai;1; ai;2).
First we recall a product decomposition of the group Sn. The algebra On has Zp *
*as its center,
hence Zxp~=Z=(p-1)xZp is central in Oxn, i.e. Zp is central in Sn. Furthermore *
*the reduced
norm which is a homomorphism from Dxnto Qxpinduces a homomorphism from Sn back *
*to Zp
which is left inverse to the inclusion of the central Zp as long as n 6 0 mod p*
*. In other words
the group Sn splits as a product Zp x S1nof the central Zp with the kernel of t*
*he reduced
norm. Following Ravenel [Ra 2] we will call this kernel S1n. Similarly the cent*
*ralizers CSn(E),
E ~=Z=p, split as CSn(E) ~=CS1n(E) x Zp, and CS1n(E) ~=Z=3 x Z3 if n = p - 1 = *
*2. The
action of the group Z=2 of Proposition 3.4 respects this splitting. In fact, Z=*
*2 acts trivially
on the central Z3 and by -1 on CS1n(E).
We need to specify the elements yi, xi, ai;1and ai;2. For this we pick a repre*
*sentative E1
of one of the two conjugacy classes and choose elements y1, x1, a1;1, a1;2of H**
*CS2(E1) as
in the remark after 3.4. If ! generates Fx9then !2 generates the group N of 3*
*.4 and acts
on H*CS2(E1) by !2(y1) = -y1, !2(x1) = -x1, !2(a1;1) = -a1;1and !2(a1;2) = a1;2.
Furthermore, ae will be linear with respect to the action of Fx9if we choose th*
*e classes y2,
x2, a2;1and a2;2such that !(y1) = y2, !(x1) = x2, !(a1;1) = a2;1and !(a1;2) = a*
*2;2. In
H*CS12(Ei) the class ai;2is missing but otherwise the same formula holds. In th*
*e discussion
below we will change notation and write ai instead of ai;1and ai0instead of ai;*
*2.
With these preparations we can finally formulate the main result of this sectio*
*n.
THEOREM 4.2. a) Let p = 3 and n = 2. ThenQthe map ae of 2.10 is a monomorphism *
*and
identifies H*S2 with the subalgebra of 2i=1F3[yi]E(xi)E(ai; ai0) generated by*
* the classes
x1, x2, y1, y2, a10+ a20, x1a1 - x2a2, y1a1 and y2a2.
b) In particular H*S2 is a finitely generated free module over F3[y1 + y2] E(a*
*10+ a20) with
8 generators which we can choose as follows: 1, x1, x2, y1, x1a1 - x2a2, y1a1,*
* y2a2 and
y1x1a1.
Before we begin with the proof we compare this calculation with the one of Rave*
*nel in [Ra 1,
Theorem 3.3]. After extension of scalars to F9 the two results have to agree. H*
*owever, they do
16
not. For example, according to Ravenel H*S2 would be multiplicatively generated*
* by classes
in degree 1 and 2 while in our computation the classes y1a1 and y2a2 are indeco*
*mposable
classes of degree 3. Furthermore, it is easy to see, that if one extends scala*
*rsQto F9 in
Ravenel's calculation, then the resulting algebra cannot be embedded into 2i=*
*1F9[yi]
E(xi) E(ai; ai0).
The two computations both give the same Poincare series, however. Furthermore b*
*oth compu-
tations give free modules over a polynomial generator of degree 2 which means t*
*hat Ravenel's
computation is not compatible with 2.10.
Because of the decomposition Sn ~=Zpx S1nit suffices to prove the following ana*
*logous result
for the group S1n.
PROPOSITION 4.3. Assume p = 3 and n = 2.
a)QThe map ae of 2.10 is a monomorphism and identifies H*S12with the subalgebra*
* of
2
i=1F3[yi] E(xi) E(ai) generated by the classes x1, x2, y1, y2, x1a1 - x2a2,*
* y1a1 and
y2a2.
b) In particular H*S12is a finitely generated free module over F3[y1 + y2] with*
* 8 generators
which we can choose as follows: 1, x1, x2, y1, x1a1 - x2a2, y1a1, y2a2 and y1x1*
*a1.
The crucial step in the proof of 4.3 is given by the following proposition.
PROPOSITION 4.4. Assume p = 3 and n = 2. There is a homomorphism S12-! Z=3
whose kernel K is torsion free and such that S12is isomorphic to the semidirect*
* product
K x||Z=3. Furthermore H*K is a Poincare duality algebra of dimension 3 and as *
*a Z=3
module H1K ~= (F3)2 is isomorphic to the augmentation ideal I(Z=3) in the group*
* algebra
F3[Z=3].
Proof_of_4.4:_____We will make use of the filtration on S1nwhich is induced fro*
*m the filtration on
Sn that we discussed in 3.1. The central Zp in Sn is topologically generated by*
* 1 + p 2 F1Sn.
Furthermore an inspection of the formula for the reduced norm (cf. [M]) shows t*
*hat it sends
FiSn onto p[i]Zp where [i] denotes the smallest integer which is bigger or equa*
*l to i, and that
it induces the trace map T r : griSn ~= Fq -! Fp ~= p[i]Zp=p[i]+1Zp if i is an *
*integer. In
particular we obtain ae
griS1n~= FqKerT r : F if i =2N
q -! Fpif i 2 N .
Furthermore the Lie bracket as well as the map P are given on grS1nby the formu*
*la of Lemma
3.1.4.
So far p and n were general. Now assume p = 3 and n = 2. As we have noted in th*
*e remark
after the proof of 3.2.1, all non-trivial elements of order 3 in S2 have the fo*
*rm 1 + aS with
a 6= 0 and a + a1+3+9 = 0, i.e a4 = -1. (Recall that a 2 F9 denotes the residu*
*e class of
a 2 O2.) In particular there is no a 2 F3 with this property. Therefore, if w*
*e identify F9
17
with (Z=3)2, and if we divide out by F3, we get a homomorphism S12-! gr1_2S12-!*
* Z=3
whose kernel K is torsionfree. Furthermore, as S12contains elements of order 3 *
*the group S12
is isomorphic to the semidirect product K x|Z=3.
The group S2 is an analytic pro 3 - group of dimension 4 and hence S12and K are*
* analytic
pro 3 - groups of dimension 3, so by [La, V.2.5.8] H*K is a Poincare duality al*
*gebra of
dimension 3. To finish it suffices to show that H1K ~=(Z=3)2 and that there are*
* elements z1
and z2 in K which project to a basis z1and z2of H1K and such that, if x 2 S12is*
* a suitable
non-trivial element of order 3, then xz1x-1 = z1z2 mod (K) and xz2x-1 = z2 mod *
*(K)
where denotes the Frattini subgroup.
Now it follows from Lemma 3.1.4 (by using [La, III.2.1.8]) that all elements in*
* F2S2 \ K are
third powers and hence H1K ~=H1Ke where eKdenotes the group K=F2S2\K. The filtr*
*ation
of S2 induces one on eK and we obtain
8 1
> KerT r : F9 -! F3 if i = 13
: F9 if i = _2
0 otherwise .
Furthermore the Lie bracket gr1_2eKx gr1Ke -! gr3_2eKand the map P : gr1_2eK-! *
*gr3_2eKare
given by [a; b] = ab3- baand P a= a+ a1+3+9. With this it is easy to check that*
* gr3_2eKis
generated by commutators and third powers and H1K ~=Ke=gr3_2eK~=(Z=3)2.
The action of an element x 2 Z=3 on H1K can now be read off from the commutator
formula in 3.1.4. Let z1 2 K and z2 2 K be elements in the appropriate filtrat*
*ion which
project nontrivially to gr1_2K resp. gr1K. The element x is represented by an*
* element
x 2 gr1_2S2 with x4 = -1. Then xz2x-1z2-1 2 K \ F3_2S2 and hence gives zero in*
* H1K.
Furthermore [x; z1] = xz31- z1x3. This is non-trivial and can be made equal to*
*_z2 if x is
chosen appropriately. In other words xz1x-1 = z1z2 mod (K) and we are done. |_|
Proof_of_4.3:_____We consider the spectral sequence of the group extension 1 ! *
*K ! S12!
Z=3 ! 1 with E2 - term E*;q2~=H*(Z=3; HqK). This is a spectral sequence of modu*
*les over
H*Z=3 and the lines q = 0 and q = 3 are free H*Z=3 - modules on one generator o*
*n the
vertical edge. Furthermore H2K ~= H1K as Z=3 - module by Poincare duality. Th*
*e exact
sequence 0 ! I(Z=3) ! F3[Z=3] ! F3 ! 0 shows that for 0 q 3 the graded vector*
* space
H*(Z=3; HqK) is additively independent of q, and in fact this is true even as m*
*odules over the
polynomial subalgebra of H*Z=3 generated by the periodicity generatorQin degree*
* 2. Therefore
the spectral sequence collapses because by 1.9 we have H*S12~= 2i=1F3[yi] E(x*
*i) E(ai)
in large degrees, and a non-trivial differential would give too small a result.
In particular we see that H*S12is a free module of rank 8 over the polynomial s*
*ubalgebra of
H*Z=3 generated in degree 2. The module generators have degree 0; 1; 1; 2; 2; 3*
*; 3; 4. By 2.10
we conclude that ae is a monomorphism, and we compute the Poincare series of th*
*e cokernel
of ae to be 1 + 2t + t2. It remains to identify the image of ae.
18
Let us first consider H1; it is two dimensional and can be identified with the *
*dual of gr1_2S12.
For each Z=3 S2 the basis element __j3in H1CS2(Z=3) (which was defined in the *
*remark
after 3.4) maps trivially to gr1_2S2. Therefore the image of ae in dimension 1 *
*is contained in
the linear span of the elements xi and by a dimension argument is equal to this*
* span.
The Bocksteins of x1 and x2 give the elements y1 and y2. However, H2S12has dim*
*ension
3,Qso we need one more element. To identify it we consider the action of Fx9on *
*H*S2 and
2
i=1F3[yi] E(xi) E(ai) (as described in 4.1) and use the linearity of ae. Th*
*e subspace
generated by y1 and y2 is invariant under this action and because the order of *
*Fx9is prime
to the characteristic we can assume that this last element is an eigenvector fo*
*r the action of
the generator ! 2 Fx9. Now there are only two eigenspaces of !, with eigenvalue*
* 1 resp. -1
and eigenvectors x1a1 + x2a2 resp. x1a1 - x2a2. By Lemma 4.5 below it is an eig*
*envector
with eigenvalue -1.
Finally ae is onto in degrees 3,_so we have determined the image of ae and the *
*missing parts of
the proposition follow easily. |_|
Remark:_______We have remarked after 3.2.2 that the map ae is not a monomorphis*
*m if n = p - 1
and p > 3. In fact, in this case we have again a decomposition Sn ~=Zp x S1nand*
* H1S1n~=
grn1_S1n~=(Z=p)n. As above one sees that for any E ~=Z=p S1nthe image of the r*
*estriction
map H1S1n-! H1CS1n(E) ~=Fp[y] E(x) E(a1; : :a:p-1) is spanned by the class x.*
* In
particular the product of any two one dimensional classes restricts trivially t*
*o all centralizers.
According to [Ra 2, Theorem 6.3.14] there are non-trivial products of one dimen*
*sional classes
as soon as p > 3 and hence ae is not injective in degree 2.
LEMMA 4.5. Let p = 3 and n = 2. The action of Fx9on H2S12~=(Z=3)3 decomposes in*
*to
a direct sum of the subspace generated by y1 and y2 with !y1 = y2, !y2 = -y1 an*
*d a one
dimensional subspace on which ! acts by multiplication with -1.
Proof:______The element ! 2 Fx9 can be lifted to a primitive 8-th root of unity*
* in W2 O2
which we will still call !. The group extension that we used in 4.4 to investi*
*gate H*S12is
not invariant under the conjugation action x 7! !x!-1 and hence not suited for *
*the problem
that we are considering here. Therefore we consider the subgroup F1S12:= S12\ F*
*1S2 which
is invariant under the action of ! and normal in S12with quotient gr1_2S12~=F9 *
*~=(Z=3)2. It
follows easily from [Lazard, V.2.2.7] that H*F1S12is an exterior algebra on 3 c*
*lasses in degree
1. We will prove the lemma by inspecting the spectral sequence
Ep;q2~=Hp((Z=3)2; HqF1S12) =) Hp+qS12
and for this we need to understand the action of (Z=3)2 on H*F1S12and the actio*
*n of Fx9on
(Z=3)2 and on H*F1S12
The element !2 is a primitive fourth root of unity for which we will write i. T*
*ogether with the
unit element it forms a basis of F9 as an F3 vector space. Therefore the elemen*
*ts a = 1 + S
and b = 1 + iS project to a basis {a; b} of gr1_2S1n, while the elements c = 1 *
*+ iS2 = 1 + 3i,
19
d = 1 + S3 and e = 1 + iS3 project to a basis {c; d; e} in H1F1S12. Furthermore*
* the relation
S! = !3S in O2 implies that the conjugation action of ! on gr1_2S12resp. on H1F*
*1S12is given
by the following formula:
!*a = -b ; !*b = a
!*c = c; !*d = -e ; !*e = d :
The action of a and bon H1F1S12can be read off from the commutator formula of 3*
*.1.4 and
we obtain:
a*(d) = d; a*(e) = e; a*(c) = c+ e
b*(d) = d; b*(e) = e; b*(c) = c- d:
With this information at hand we can look at the spectral sequence. The importa*
*nt groups for
us are E1;12and E3;02as modules over Fx9. A straightforward computation gives E*
*1;12~=(Z=3)3
and E3;02~=(Z=3)4. Furthermore with respect to the !-action E1;12decomposes as *
*a sum of
a two-dimensional eigenspace with eigenvalue -1 and a one-dimensional eigenspac*
*e with
eigenvalue 1 while E3;02decomposes into a direct sum of two two-dimensional eig*
*enspaces
with respective eigenvalues 1 and -1.
From the proof of 4.3 we know already that the classes x1 and x2 are represente*
*d on E1;02
and consequently the classes y1 and y2 are represented on E2;02. Because we kn*
*ow already
that H2S1nis of dimension 3 it follows that the kernel of the differential d2 :*
* E1;12-! E3;02is
at most one-dimensional, and because we also know that the classes x1y1 and x2y*
*2 in E3;02
survive to E1 the kernel is precisely one dimensional and gives the missing cl*
*ass in H2S12.
We have to show that this kernel is contained in the -1 eigenspace of !.
In fact, E3;03, the quotient of E3;02by the image of d2 is generated by x1y1 an*
*d x2y2 and is
a direct sum of two one-dimensional eigenspaces with eigenvalues 1 resp. -1. Th*
*e decom-
position of E3;02implies that the image of d2 is also a direct sum of two one-d*
*imensional
eigenspaces with eigenvalues 1 resp. -1, and hence the_decomposition of E1;12gi*
*ves that !
acts by multiplication by -1 on the kernel of d2. |_|
5. The case GL(n; Zp)
5.1.______We start with a few general remarks on the continuous cohomology of t*
*he groups
GL(n; Zp). Mod pr - reduction defines maps from GL(n; Zp) to GL(n; Z=pr) with *
*kernel
(pr). The groups (pr) form a decreasing sequence of closed subgroups with GL(n;*
* Zp) ~=
lim GL(n; Zp)=(pr). Furthermore, the quotients (pr)=(pr+1) are elementary abel*
*ian p -
groups (of rank n2) and hence (p) ~=lim(p)=(pr) is a profinite p - group.
In particular, for a prime l 6= p, mod p - reduction induces an isomorphism in *
*continuous co-
homology H*(GL(n; Z=p); Fl) -! H*(GL(n; Zp); Fl) and hence H*(GL(n; Zp); Fl) is*
* known
by the work of Quillen [Q2]. However, if p = l, then very little seems to be k*
*nown about
H*(GL(n; Zp); Fp) = H*GL(n; Zp) (from now on we will omit the coefficients agai*
*n in our
20
notation). For example, studying mod p - reduction does not lead very far, beca*
*use the mod
- p - cohomology of the quotient GL(n; Fp) is not known unless n is very small.
We will use our centralizer approach to compute H*GL(n; Zp) in large dimensions*
* in case
n = p - 1, p odd. The following result gives the necessary group theoretic inf*
*ormation to
apply Theorem 1.4 resp. Corollary 1.7.
THEOREM 5.2. Let p be odd and n = p - 1.
a) The p - rank of GL(n; Zp) is equal to one and, up to conjugacy, there is a u*
*nique subgroup
E of GL(n; Zp) which is isomorphic to Z=p.
b) The centralizer CGL(n;Zp)(E) is isomorphic to Zp[ip]x ~=Z=p x (Zp)n x Z=(p -*
* 1).
c) Aut A*(GL(n;Zp))(E) ~= Z=n and the action of Z=n on CGL(n;Zp)(E) corresponds*
* via the
isomorphism of (b) to the Galois action on Zp[ip]x which was explicitly describ*
*ed in 3.4.
The crucial input for 5.2 is the following p-adic version of the theorem of Die*
*derichsen and
Reiner [CR, Theorem (74.3)]. It can be proved in the same way as the integral v*
*ersion except
that some of the details simplify because class group phenomena dissappear in t*
*he p - adic
version.
THEOREM 5.3. Let G = Z=p and M a Zp[G] - module which is finitely generated and*
* free as
Zp - module. Let F = Zp[G] be the free Zp[G]- module on one generator, T ~=Zp t*
*he trivial
one dimensional module, and R = Zp[ip] the ring of integers in the cyclotomic e*
*xtension
Qp(ip) with action of a generator g 2 G given by gr = ipr for r 2 R.
Then
M ~=F k T l Rm
_
for a unique triple (k; l; m) of non-negative numbers. |_|
Proof_of_5.2:_____a) The Zp[G] - module R is isomorphic to (Zp)n as Zp - module*
* which shows
that there is an embedding of G into GL(n; Zp). By Theorem 5.2 there is a uniqu*
*e Zp[G] -
module structure on (Zp)n for which the action is faithful which means that all*
* subgroups E
of order p in GL(n; Zp) are conjugate. That the p - rank is not bigger than 1 w*
*ill follow from
part b) because the p - rank of the centralizer of E is only 1.
b) We have isomorphisms CGL(n;Zp)(E) ~=AutZp[E](R) and because the Zp[E] module*
* struc-
ture on the ring R is pulled back from the R - module structure we also get Aut*
*Zp[E](R) ~=
Aut R(R) ~=Rx .
c) The group AutA*(GL(n;Zp))(E) is a subgroup of the group of all abstract auto*
*morphisms of
E and the Galois_group of the cyclotomic extension realizes all of them through*
* conjugations
in GL(n; Zp). |_|
Using the notation of 3.3 and 3.4 we write H*CGL(n;Zp)(E) ~=Fp[y] E(x) E(a1; *
*. .;.an).
The action of the Galois group is determined by 3.4. Combining Theorem 5.2 with*
* Corollary
1.7 leads to the following result (Theorem 1.10 of the introduction).
21
THEOREM 5.4. Let p be an odd prime, n = p - 1 and E ~= Z=p GL(n; Zp). Then t*
*he
restriction map
ae : H*GL(n; Zp) -! (H*CGL(n;Zp)(E))Z=n ~=(Fp[y] E(x) E(a1; . .;.an))Z=n
_
has finite kernel and cokernel. |_|
As in [A] one can analyze all p - rank one cases p - 1 n 2p - 3 further and r*
*educe the
computation of H*GL(n; Zp) in large dimensions to the computation of the cohomo*
*logy of
GL(n - p + 1; Zp) and of appropriate congruence subgroups thereof. We leave the*
* details to
the interested reader.
5.5.______We finish with a brief discussion of the case p = 3 and n = 2. This c*
*ase is simple enough
that one could do it directly with standard methods. However, we include it her*
*e as another
example illustrating our theory and how the map ae of 1.7 may fail to be an iso*
*morphism in
small dimensions.
The situation is very similar to that of the group S2 for the prime 3 that we d*
*iscussed in section
4. Using the same notation as in 4.1 we write H*CGL(2;Z3)(Z=3) ~=Fp[y] E(x) E*
*(a; a0)
where y is of degree 2 and all the other classes are of degree 1 and the action*
* of the non-trivial
element g in the Galois group is trivial on a0 and multiplies all other generat*
*ors by -1. In
particular, the ring of invariants (F3[y]E(x)E(a; a0))Z=2 is equal to the subri*
*ng generated
by the elements y2; yx; ya; xa and a0. This is a free module of rank 4 over F3[*
*y2] E(a0) on
generators 1; yx; ya and xa.
PROPOSITION 5.5. The restriction map ae : H*GL(2; Z3) -! (H*CGL(2;Z3)(Z=3))Z=2*
* ~=
(F3[y]E(x)E(a; a0))Z=2 is a monomorphism and identifies H*GL(2; Z3) with the su*
*balgebra
F3[y2] E(yx; ya) E(a0) of the invariants.
Proof:______First we note that GL(n; Zp) is isomorphic to Zp x GL1(n; Zp) where*
* GL1(n; Zp) de-
notes the subgroup of GL(n; Zp) which is the preimage of Z=p-1 Zxpunder the de*
*terminant
map. In fact, Zxp~=Zp x Z=p - 1 identifies with the center of GL(n; Zp) and the*
* composition
with the determinant is multiplication by n, hence an isomorphism on the Zp sum*
*mand of
Zxpif n 6 0 mod p. Furthermore, H*GL1(n; Zp) ~=(H*SL(n; Zp))Z=p-1,_and hence 5.*
*5 will
follow from the following result for the special linear group. |_|
First note that if one works with SL(2; Z3) instead of GL(2; Z3) then one still*
* has a unique
subgroup Z=3 up to conjugacy with centralizer CSL(2;Z3)~=Z=2xZ=3xZ3 and H*CSL(2*
*;Z3)(E)
~=F3[y] E(x) E(a). Furthermore, in this case AutA*(SL(2;Z3))(Z=3) is the triv*
*ial group so
that in large degrees H*SL(2; Z3) ~=F3[y] E(x) E(a) by 1.7.
PROPOSITION 5.6. The restriction map ae : H*SL(2; Z3) -! H*CSL(2;Z3)(Z=3)) ~=F3*
*[y]
E(x; a) is a monomorphism and identifies H*SL(2; Z3) with F3[y] E(x; ya).
22
Proof:______We consider the mod 3 reduction map SL(2; Z3) -! SL(2; F3). The ke*
*rnel K is
a torsion free 3 - dimensional analytic pro 3 - group and hence H*K is a 3 - di*
*mensional
Poincare duality algebra by [La,V.2.5.8]. Now consider the graded Lie algebra a*
*ssociated to
the decreasing filtration of SL(2; Z3) by the kernels of mod 3k reduction, k = *
*1; 2; : :.:It is
easy to see from this Lie algebra, say as in the proof of 4.4, that H1K ~=(F3)3*
*. Furthermore,
using [La,V.2.2.7] we see that H*K is exterior on the 3 generators in degree 1.
Now we consider the spectral sequence of the extension
1 -! K -! SL(2; Z3) -! SL(2; F3) -! 1
with E*;q2~=H*(SL(2; F3); HqK). The group SL(2; F3) acts necessarily trivial o*
*n H3K
because there are no non-trivial homomomorphisms from SL(2; F3) to GL(1; F3). N*
*ext one
can check that the invariants of H1K with respect to the action of the 2 - Sylo*
*w subgroup Q8 of
SL2(F3) are trivial. Because Q8 is normal in SL(2; F3) this implies H*(SL(2; F3*
*); H1K) = 0
and by Poincare duality we also have H*(SL(2; F3); H2K) = 0. Finally the restri*
*ction map to
the 3 - Sylow subgroup is well known to induce an isomorphism H*SL(2; F3) ~=H*Z*
*=3 and so
our spectral sequencs has just 2 non-trivial rows at E2, which are both isomorp*
*hic to H*Z=3.
As in the proof of 4.3 we see now that the spectral sequence has to collapse; a*
* non-trivial
differential would lead to a result which is too small to be compatible with 1.*
*7. Then the
spectral sequence shows that H*SL(2; Z3) is free over F3[y] and hence ae is inj*
*ective. It is clear
that the elements x and y are in the image of ae and by a counting dimensions o*
*ne sees that ae_
is an isomorphism in degrees 3 and bigger, in particular the image of ae is F3[*
*y] E(x; ya). |_|
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Hans-Werner Henn
Mathematisches Institut der Universit"at
Im Neuenheimer Feld 288
D-69120 Heidelberg
Fed. Rep. of Germany
Current address:
Max Planck Institut f"ur Mathematik
Gottfried Claren Strasse 26
D-53225 Bonn
Fed. Rep. of Germany
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